A sequence is an ordered list of numbers where each number is called a term. The first term is denoted as a₁. In our example, a₁ equals 2. Sequences can follow specific patterns or rules that determine how each term relates to the previous ones.
Sequences can follow different patterns. Let's explore three types starting with a₁ = 2. First, arithmetic sequences have a constant difference between consecutive terms. Here we add 2 each time. Second, geometric sequences have a constant ratio between consecutive terms. Here we multiply by 2 each time. Third, there are other patterns where the differences themselves follow a rule.
In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. The general formula is aₙ equals a₁ plus (n minus 1) times d, where d is the common difference. With a₁ equals 2 and d equals 3, we get aₙ equals 2 plus (n minus 1) times 3. Let's calculate the first few terms: a₁ equals 2, a₂ equals 2 plus 3 equals 5, a₃ equals 5 plus 3 equals 8, and so on.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. The general formula is aₙ equals a₁ times r to the power of (n minus 1), where r is the common ratio. With a₁ equals 2 and r equals 2, we get aₙ equals 2 times 2 to the power of (n minus 1). Let's calculate the first few terms: a₁ equals 2, a₂ equals 2 times 2 equals 4, a₃ equals 4 times 2 equals 8, and so on. This creates an exponential growth pattern.
A recursive sequence defines each term based on one or more previous terms. Instead of having a direct formula for aₙ, we define the first term and then give a rule to find any term from previous terms. For example, with a₁ equals 2 and aₙ equals aₙ minus 1 plus 3, we start with 2 and add 3 to get each next term. So a₂ equals a₁ plus 3 equals 2 plus 3 equals 5, a₃ equals a₂ plus 3 equals 5 plus 3 equals 8, and so on.
In Fibonacci-type sequences, each term is the sum of the two preceding terms. This is a second-order recurrence relation. Let's start with a₁ equals 2 and a₂ equals 1, different from the classic Fibonacci sequence. Then a₃ equals a₂ plus a₁ equals 1 plus 2 equals 3, a₄ equals a₃ plus a₂ equals 3 plus 1 equals 4, a₅ equals a₄ plus a₃ equals 4 plus 3 equals 7, and so on. These sequences often appear in nature and have interesting mathematical properties.
A sequence converges if its terms approach a specific value as n increases. This value is called the limit. Let's examine three cases starting with a₁ equals 2. First, the sequence aₙ equals 2 plus 1 over n converges to 2. As n gets larger, 1 over n approaches 0, so aₙ approaches 2. Second, the sequence aₙ equals 2 to the power of n diverges to infinity. The terms grow without bound. Third, the sequence aₙ equals 2 plus negative 1 to the power of n divided by n also converges to 2, but oscillates around the limit as it approaches it.
Sequences have many real-world applications. Let's look at compound interest as an example. If you invest 2 dollars at 100 percent annual interest, compounded annually, the amount grows geometrically. After 0 years, you have 2 dollars. After 1 year, you have 2 times 1 plus 1 equals 4 dollars. After 2 years, you have 4 times 2 equals 8 dollars. This follows the formula A equals P times (1 plus r) to the power of t, where P is the principal, r is the rate, and t is time. Sequences also model population growth, algorithm complexity, and many other phenomena.